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Homoclinic Cycle Bifurcations in Planar Maps

Kyohei Kamiyama

Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan

E-mail Address: kamiyama@sat.t.u-tokyo.ac.jp

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Motomasa Komuro

Center for Fundamental Education, Teikyo University of Science, 2525 Yatsusawa, Uenohara-shi, Yamanashi 409-0193, Japan

E-mail Address: komuro@ntu.ac.jp

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Kazuyuki Aihara

Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan

E-mail Address: aihara@sat.t.u-tokyo.ac.jp

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https://doi.org/10.1142/S0218127417300129Cited by:0 (Source: Crossref)

Abstract

In this study, bifurcations of an invariant closed curve (ICC) generated from a homoclinic connection of a saddle fixed point are analyzed in a planar map. Such bifurcations are called homoclinic cycle (HCC) bifurcations of the saddle fixed point. We examine the HCC bifurcation structure and the properties of the generated ICC. A planar map that can accurately control the stable and unstable manifolds of the saddle fixed point is designed for this analysis and the results indicate that the HCC bifurcation depends upon a product of two eigenvalues of the saddle fixed point, and the generated ICC is a chaotic attractor with a positive Lyapunov exponent.

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